Height bounds and the Siegel property

Let G be a reductive group defined over Q and let S be a Siegel set in G(R). The Siegel property tells us that there are only finitely many gamma∈G(Q) of bounded determinant and denominator for which the translate gamma.S intersects S. We prove a bound for the height of these gamma which is polynomial with respect to the determinant and denominator. The bound generalises a result of Habegger and Pila dealing with the case of GL2, and has applications to the Zilber–Pink conjecture on unlikely intersections in Shimura varieties.

In addition we prove that if H is a subset of G, then every Siegel set for H is contained in a finite union of G(Q)-translates of a Siegel set for G.